Can you picture the circle on the Argand diagram? If so, then you should be able to see at which point on the circle where z is a 'vector' from the origin of the circle to a point on the circumference, would result in the greatest angle arg(z).

EDIT: I think a better word to use, as Dick has said, would be "tangent". z is tangent to the circle.

arg(z) is just the angle that a line through z and the origin makes with the x axis. The z that makes the maximum value of that angle must be the point of intersection with the circle of a line tangent to the circle starting at the origin. Seems to me a clever person could draw some right triangles and make a trig problem out of this.

ok so I drew the circle, see the the real axis is a tangent to the circle at (2,0) and the Im(z) intersects the circle. So the largest arg(z) would be the tangent to the circle at the part where the circle intersects the axis?

The center of the circle is (1,2). It has radius 2. I don't think it's tangent to the real axis at (2,0). It's in your best interests to draw an accurate picture or you'll waste a lot of time. The z having largest arg(z) doesn't have to be on any axis. It isn't.

ok so I drew the circle, see the the real axis is a tangent to the circle at (2,0) and the Im(z) intersects the circle. So the largest arg(z) would be the tangent to the circle at the part where the circle intersects the axis?